Competition problem — Combinatorics

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Combinatorics Difficulty 4.9 AIME Prove it United States

Problem:

Louis moves around on the lattice points according to the following rules: From point $(x, y)$ he may move to any of the points $(y, x)$ , $(3x, -4y)$ , $(-2x, 5y)$ , $(x+1, y+6)$ and $(x-7, y)$ . Show that if he starts at $(0,1)$ he can never get to $(0,0)$ .

Solution

Solution:

Call a point $(x, y)$ stable if $x + y$ is not divisible by $7$ . The key is to observe that starting from a stable point, one may only reach other stable points. For example, $3x - 4y \equiv 3(x + y) \pmod{7}$ , hence if $(x, y)$ is stable then $(3x, -4y)$ is as well.
Consequently, starting from the stable point $(0,1)$ it's impossible to reach the unstable point $(0,0)$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.